### Index Notation Partials and Rotations, a Cool Gradient Trick: EMII Notes 2014_08_09 Part I

Summary of what's gone on before.  Finally got the Levi-Civita to Kronecker delta identity down yesterday, 2014/08/08.  Today we're making more use of the index notation.  There are however, some notational stumbling points

$r^2 = x^2 + y^2 + z^2$

$\vec{r} = \left(x, y, z\right)$

Now if we want to take the derivative of both sides of the magnitude equation above, first remember we can write $r^2$ as

$r^2 = x_jx_j$

Now, finally taking the derivative of both sides o the above we get

$2r \partial_i r = 2x_j\partial_i x_j$

remembering the partial differentiation rules and the kronecker delta we can write the above down as

$2r \partial_i r = 2x_j \delta_{ij} = 2x_i$

which finally gives:

$\partial_i r = \dfrac{x_i}{r}$

The Kronecker trick above is crucial. Also remember, not one of the r's is a vector, they're all the magnitude of the vector.

\section{Rotations}

Any rigid rotation of a vector can be defined as:

$\begin{pmatrix} x'\\ y'\\ z' \end{pmatrix} = M\begin{pmatrix} x'\\ y'\\ z' \end{pmatrix}$

Here are a few of the important parts.  The Matrix $M$ is orthogonal and, $M^TM = 1$

The transpose actually defines orthogonality.  If such a matrix is of dimension n, then it is n $)\left(n\right)$ matrix.

The footnote has all the cool kid stuff about rotation matrices and how to name them:

First of all, if the determinant of a matrix is +1, then it's a special orthogonal matrix, $SO\left(n\right)$.  The other sort, the sort with a determinant of -1 are actually rotations with a reflection of the coordinates.

There are also some identities in the footnote that will come in handy

$det\left(AB\right) = \left(det A\right)\left(B\right)$

and

$det\left(A^T\right) = det\left(A\right)$

Using the above two identities, we can see that $\left(det M\right)^2 = 1$

Memorize these!!!

In index notation, the rotation above can be written as
$x_i^{\prime} = M_{ij}x_j$

The orthogonality condition becomes

$M_{ki}M_{kj} = \delta_{ij}$

### Cool Math Tricks: Deriving the Divergence, (Del or Nabla) into New (Cylindrical) Coordinate Systems

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The following is a pretty lengthy procedure, but converting the divergence, (nabla, del) operator between coordinate systems comes up pretty often. While there are tables for converting between common coordinate systems, there seem to be fewer explanations of the procedure for deriving the conversion, so here goes!

What do we actually want?

To convert the Cartesian nabla

to the nabla for another coordinate system, say… cylindrical coordinates.

What we’ll need:

1. The Cartesian Nabla:

2. A set of equations relating the Cartesian coordinates to cylindrical coordinates:

3. A set of equations relating the Cartesian basis vectors to the basis vectors of the new coordinate system:

How to do it:

Use the chain rule for differentiation to convert the derivatives with respect to the Cartesian variables to derivatives with respect to the cylindrical variables.

The chain rule can be used to convert a differe…

### Division: Distributing the Work

Our unschooling math comes in bits and pieces.  The oldest kid here, seven year-old No. 1 loves math problems, so math moves along pretty fast for her.  Here’s how she arrived at the distributive property recently.  Tldr; it came about only because she needed it.
“Give me a math problem!” No. 1 asked Mom-person.

“OK, what’s 18 divided by 2?  But, you’re going to have to do it as you walk.  You and Dad need to head out.”

And so, No. 1 and I found ourselves headed out on our mini-adventure with a new math problem to discuss.

One looked at the ceiling of the library lost in thought as we walked.  She glanced down at her fingers for a moment.  “Is it six?”

“I don’t know, let’s see,” I hedged.  “What’s two times six?  Is it eighteen?”

One looked at me hopefully heading back into her mental math.

I needed to visit the restroom before we left, so I hurried her calculation along.  “What’s two times five?”

I got a grin, and another look indicating she was thinking about that one.

I flashed eac…